Winning Lottery with Prime Number!?

[jimfan]

Yesterday night we had a lottery snow ball more than HKD 110 million.

Suddenly an idea floated up: Will betting all prime numbers get me first prize?

The draw picks 6 plus 1 special number out of 1 to 49. A gambler bets for 6 out of 1 to 49. If gambler hit all 6 non-special number, he gets first prize.

Result: 6 out of 7 drawn numbers are prime!

The first prize is made up of 5 prime number.

I got 4 of them. Prize is HKD 640, around USD 80

Quite a mysterious experience.

If you are interested in the drawn numbers, see http://bet.hkjc.com/marksix/?lang=ch

[jaap]

There are 15 primes in the range [1,49].
There are 15C7 + 34C1*15C6 = 176605 ways to choose at least 6 of those primes in a drawing of 7 balls.
There are 49C7 = 85900584 ways to draw 7 balls.
This gives a probability of 176605/85900584 = 0.00205.., which is about 1 in 500.
So this is a rare occurrence, but it is not that improbable for it to occur when there are regular draws in a long-running lottery.

[jimfan]

Hi jaap,

Thanks for demonstrating the probability!

Only the idea of "betting prime numbers" didn't pop up in my mind at all before, together with the 6/7 prime, the coincidence adds a little mistique... At least from my oriental perspective

I am betting on Fibonacci numbers in coming draw, which is not a very clever choice. As Fibonacci numbers within 1-49 are small, the likelihood of having to share first prize is statistically higher (according to a British book I browsed in library).

Reason: "Birthday numbers" (1-31) are more popular in bets.

From my non-mathematical observation, this is true. And this is why in the prime number bet, I started picking at 29.

Jim

[Gimmesumcredit]

Hi Jim,

Do you have details on the book you mentioned?

I've done some work in this area. Here's a link to a talk I gave on my findings.

https://www.youtube.com/watch?v=vNZRSYWOJBE&t=25s


Regards,
Stephen

[jimfan]

Hello Stephen,

Thank you for sharing your presentation. It seems that "birthday numbers are more popular" applies to the West too.

Unluckily I could not recall the book correctly. At first I thought the book was "What Are the Odds?" by Tim Glynne-Jones, published by Arcturus Publishing Limited (ISBN: 978-1-84837-853-7). Browsed again but couldn't find the statement.

Yet I firmly believe the idea is from casual browsing of math column in library - after all I am not clever enough to come up with the idea myself...

In 16:45 of your presentation you have made the same statement I've read: Betting on big numbers won't win more easily; Only the opportunity of having to share the prize is lower.

On the other hand, it appears to me US lottery organisers are more open with their data. I couldn't expect local organiser releasing bet data as such.

Jim

[sjhillier]

Interesting talk. For fun, some years ago (20ish!) I did a very naive analysis of the UK lottery to try to determine popular numbers. My results were not very strong, but there was certainly some indication of 1-30 being more popular. However, I noticed another trend that might be worth exploring.

At the time, the lottery slips were laid out in a grid of 5x10 numbers (1-5, 6-10, etc) up to 49. (It's a 6/49 lottery.) I'm not sure if the slips are still the same, as I haven't looked at them for about 20 years. I got the impression that the numbers at the edge of the grid were less popular, and in particular corner numbers, such as 46 and 49. This doesn't apply to the lower numbers which seem to be more popular anyway. But 46 and 49 seemed particularly unpopular, and to a lesser extent 31, 35, 36, 40, 41, 45, 47, and 48. Even amongst the numbers in the range 1-30, the side numbers were possibly marginally less popular. My assumption would be that people filling in a lottery choice without any prior ideas about what numbers to choose tend to gravitate towards the interior of the grid, and pick numbers on the edge of the grid less often. Would be interesting to see if this is still the case, but I don't have the time for that sort of thing nowadays!

[RobertStanforth]

I believe the book that jimfan refers to is "From Here to Infinity" by Ian Stewart.

[Roquentin]

(according to a British book I browsed in library).

Sounds interesting. Do you recall what the title was? Was this book only about the lottery, or statistics as a whole?

[jimfan]

Possibly be "The Call of the Primes: Surprising Patterns, Peculiar Puzzles, and Other Marvels of Mathematics", by Owen O'Shea, publisher Prometheus Books. If I am correct both are from British.